| Summary: | We examine the discrete Ponzano-Regge formulation of (2+1)-dimensional gravity in
the context of a consistent histories approach to quantum cosmology. We consider 2-
dimensional boundaries of a 3-dimensional spacetime. The 2-dimensional boundaries are
tessellated as the surface of a single tetrahedron. Two classes of the tetrahedral tessellation
are considered—the completely isotropic tetrahedron and the two-parameter
anisotropic tetrahedron. Using Ponzano-Regge wavefunctions, we calculate expectation
values and uncertainties for the edge lengths of these tetrahedra. In doing so, we observe
finite size effects in the expectation values and uncertainties when the calculations
fail to constrain the space of histories accessible to the system. There is, however, an
indication that the geometries of the tetrahedra (as quantified by the ratios of their edge
lengths) freeze out with increasing cutoff. Conversely, cutoff invariance is observed in
our calculations provided the space of histories is constrained by an appropriate fixing
of the tetrahedral edge lengths. It is thus suggested that physically meaningful results
regarding the early state of our universe can be obtained providing we formulate the
problem in a careful manner. A few of the difficulties inherent in quantum cosmology
are thereby addressed in this study of an exactly calculable theory.
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